The fundamental perturbation theorems required by (II) are provided by Fenichel theory, which is constituted by a collection of perturbation and invariant manifold theorems derived in [1]}, [2]}, [3]} and culminating in [4]}
(though it is also important to mention the earlier works of [5]}, [6]}, which provided foundational understanding for the field). Fenichel theory ensures that suitably constructed singular orbits perturb in a regular fashion for \(0 < \varepsilon \ll 1\)  in regions of phase space bounded away from certain singularities on \(S\) . Fenichel theory applies locally near normally hyperbolic submanifolds of \(S\)  for which the linearization along the (\(k-\) dimensional) critical manifold \(S\)  has \(n-k\)  non-trivial eigenvalues bounded away from the imaginary axis. For fast-slow systems (REF ), this is equivalent to the requirement that the \((n-k)\times (n-k)\)  square matrix \(DfN|_S\)  are bounded away from the imaginary axis [7]}. Under normally hyperbolic conditions, the geometry and dynamics for \(0 < \varepsilon \ll 1\)  are described up to \(O(\varepsilon )\)  accuracy by the dynamics of layer and reduced problems, which are almost invariably far more tractable analytically.
Additional perturbation theorems are required in order to describe the geometry and dynamics near non-normally hyperbolic submanifolds of \(S\) . Many authors have demonstrated the utility of a method of geometric desingularization known as the blow-up method for such purposes; here we simply cite the seminal works in
[8]}, [9]} and refer to the recent survey [10]}.
This combination of GSPT and blow-up techniques has been applied by many authors [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}.
